Microeconomics MSc - Haifatodd/beamerauctionshandout.pdf · Microeconomics MSc Auctions Todd R. Kaplan University of Haifa December 2009 ... One™s bid does not a⁄ect the price

Microeconomics MScAuctions

Todd R. Kaplan

University of Haifa

December 2009

Kaplan (Haifa) micro December 2009 1 / 24

Auctions

An auction is a mechanism for trading items by means of bidding(o¤ering prices).

Dates back to 500 BC where Babylonians auctioned of women aswives.

Position of Emperor of Rome was auctioned o¤ in 193 ad.

Recently auctions were brought into both everyday use with theInternet (ebay and Google) and governmental arena with thespectrum auctions, bonds and environmental markets.

Can have the bidders trying to buy an item: Christie�s, ebay.

Can have the bidders trying to sell an item : Procurement,priceline.com.


Rules to Auctions

First-Price Sealed-Bid Auction: Everyone writes down a bid insecret. The person with the highest bid wins the object and payswhat he bids.

Second-Price Sealed-Bid (Vickrey) Auction: Everyone writesdown a bid in secret. The person with the highest bid wins the objectand pays the second highest bid. (used for stamps and by Goethe).

English Auction: The auctioneer starts at a reserve price andincreases the price until only one bidder is left.

Dutch Auction (Not Demonstrated): The auctioneer starts at a highprice and decreases the price until a bidder accepts the price. (Similarto price-drop.tv)

All Pay Auction: Everyone writes down a bid in secret. The personwith the highest bid wins. Everyone pays


Tokyo Fish Auction


Tokyo Flower Auction


Classroom Experiment

You will be in teams. (Probably 5 teams.)

Each team will have an independent value for winning drawnuniformly from [0,10].

Write down your bid in each format: First-Price, Second-Price, andAll-pay.

Be careful that you are not overheard by other teams.

I will collect the bids and tabulate results.

I will chose at random one of the auction formats to count eachperiod. I will collect the payments according to the rules and pay thewinning team�s value in NIS.

Ties will be broken randomly.

This will be repeated several times.


Two types of Settings: Common and Private

1 Examples of Common Auctions:

Spectrum.Oil Drilling.

2 Examples of Private Auctions:

Consumption items.Memorabilia


Strategies with Private values: English Auction

The English: stay in the auction until either

you winor the bid goes higher than your value.

With a di¤erent strategy:

either one loses when it is worthwhile to winone wins when it is worthwhile to lose.

The key to understanding this is to understand that staying in doesnot a¤ect the price one pays if they win only whether one wins (itdoes a¤ect others�prices).

It is best to do this independent of what others do.


Strategies with Private Values: 2nd Price Auctions

2nd price similar logic to English auction.

It is optimal to bid one�s value.

One�s bid does not a¤ect the price one pays only whether or not onepays.

Raising one�s bid will cause one to win when it is not worthwhile.

Your value is 5, the other team bids 6. If you bid 7, you will lose:5-6=-1.

Lowering one�s bid will cause one to lose when it was worthwhile towin.

Your value is 5, the other team bids 4. If you bid 3, you won�t winand could have won 5-4=1.

Again it is best to do this independent of what others do.


Example of Strategy

Values are uniform [0, 10] (equal chance of each). Team B bids itsvalue Vb. Team A receives value Va=5 and has to decide on a bid

Team A bids its value minus 1. In this case, Team A wins wheneverVb < 4 and pays B�s value. Expected pro�t is 4/10 � (5� 2) = 1.2Team A bids its value. In this case, Team A wins whenever Vb < 5and pays B�s value. Expected pro�t is 5/10 � (5� 2.5) = 1.25.Team A bids its value plus 1. In this case, Team A wins wheneverVb < 6 and pays B�s value. Expected pro�t is 6/10 � (5� 3) = 1.2.Problem in 1st case, A doesn�t win when 4 < Vb < 5. Problem in2nd case, A wins when 5 < Vb < 6.


Strategies with Private Values: First Price

Strategies in the �rst-price should shade bid below your value

This is because one�s bid a¤ects one�s price.Bidding your value will earn zero surplus.Shading one�s bid lowers the probability of winning, but increases thesurplus gained when winning.

There is a natural trade-o¤ between probability of winning and pro�tif one wins.

If bid is b, value is v , expected pro�t is Probwin(b)(v � b)Derivative of this w.r.t. b yields Probwin0(b)(v � b)� Probwin(b) = 0First term is marginal bene�t of higher prob of winning.Second term is marginal cost to the pro�t. (chance of winning timecost if one wins)


First-Price Auction w/ Uniform Distribution

In FP auction the best strategy depends upon what others do (theya¤ect probwin).

If everyone bids zero, then you want to bid close to zero.If everyone bids above 1, you don�t want to bid close to zero if yourvalue is 2.

Equilibrium is

all bidders choose a strategy b(v) : how much to bid given their value.Given the strategy of others, no one wants to change their strategy(Nash).


First price auction equilibrium

b�(v) is an equilibrium such that

b�(v) = argmaxbprobwin(b)(v � b)

What is prob of winning given there are only 2 bidders and the otherbidder bids b�(v)?

Prob(b�(v) < b) = Prob(v < b�1(b))Easiest to see this graphically.What is this prob. if b�(v) = av and F (v) = v (uniform)?What happens if there are N � 1 other bidders?


FP auction equilibrium

Prob of winning given bid b and N � 1 others are bidding accordingto a � v is

� ba

�N�1A bidder maximizes maxb ba

(N�1)(v � b)

For N = 2, we have the �rst-order condition of . . . . . . . . . . . . . . . .(Independent of a).

This is maximized at b = v2 .

Exercise, do this for N = 3.

Anyone want to guess the formula for any N?


Strategies with Private Values: All Pay

In the all-pay auction, you should again shade bid below your value.

If bid is b, value is v , expected pro�t isProbwin(b)v � b. The natural trade-o¤ is now between probability ofwinning and cost of bidding.

This cost is incurred whether you win or not.

Derivative of expected pro�ts w.r.t. b yieldsProbwin0(b)v � 1 = 0First term is marginal bene�t of higher prob of winning.Second term is marginal cost of increased bid.

It only makes sense to incur a high cost if the probability of winning isfairly high.

For low values, bids are shaded much more than with �rst-priceauctions.


All-pay equilibria: Linear soln?

Prob of winning given bid b and N � 1 others are bidding accordingto a � v is (b/a)N�1

A bidder maximizes Maxb( ba )N�1v � b

For N = 2, we have a clear problem Maxb(b/a)v � bIt is a corner solution unless a = v , which is a contradiction (a isconstant and v varies).

We instead need a slightly di¤erent approach.


All-pay auction: equilibrium

Before the bidder chose a bid given his value.

Probability of winning was the chance that the other bidder had avalue that would induce a lower bid.

Now have the bidder choose a �pretend�value ev given their true valuev . Their bid is then b(ev).Bidders then maxev evv� b(ev)An equilibrium would just have the bidder choose ev = v . Thus,b0(v) = v .

Integration yields b(v) = v 22 + const. The constant is 0, since

someone having v = 0 would not to bid.

What is it for N > 2 bidders?


All-pay auction

May seem like a strange auction to run/study (one which I doresearch), but. . .

It is used in charity auctions and from the lab, one can see why.(Losers don�t complain so much.)

Extremely useful modelling tool.

Patent Races.Political Campaigns.Technology contests �X-prize, Lindbergh.Procurement contests �Architecture, Next Generation Fighter Jet.Sports contests. (Think of Chelsea, Man U, Arsenal all buying the bestplayers.)


Summary: Strategies with uniform values.

Values are drawn from 0 to 10 with an equal chance of each amount(like in the experiment). N is the number of bidders.

1st-price the equilibrium bid N�1N v (that is if v = $5.50 and N = 2,

bid $2.75.

Dutch auction is the same as the 1st-price.

2nd-price, optimal to bid value. English optimal to bid up to one�svalue.

All-pay auction, should bid (N � 1) �� v10

�N � 10N (looks complicated

but only we can see for low values shade bid more than for highvalues).


Equilibrium Bid Functions

2 4 6 8 10

2

4

6

8

10

2nd PriceBid

Value

1stprice

AllPay


Which design is best for revenue?

Basic probability.

Take N values independently drawn from the uniform distribution[0, 1].

N = 1, the expected value is 1/2.

N = 2, the expected highest maxfv1, v2g = 2/3, the expected lowestminfv1, v2g = 1/3.Maxfv1, . . . , vNg = N/(N + 1).Minfv1, . . . ., vNg = 1/(N + 1).The expected value of the 2nd highest= (N � 1)/(N + 1)For a uniform distribution, [0, a], E [Maxfv1, . . . , vNg] = N

N+1 � a, etc..


Expected Revenue

Everyone bids their value. The highest wins it and pays the 2ndhighest bid. What is the 2nd highest value?

N�1N+1 . For N = 2, it is just 1/3.

For a �rst-price auction, the highest wins it and pays his bid. He bidsN�1N � v .E [N�1N maxfvig] = N�1

N � E [maxfvig] =N�1N � N

N+1 =N�1N+1 . Same!

In all-pay, the auctioneer collects money from all bidders.Revenue=N

R 10 b(v)dv = N

R 10N�1N vNdv = N�1

N+1

They are all the same! Amazing!


Revenue Equivalence: 4 designs

For private values, there is revenue equivalence among all 4 designs:1st/2nd price, all-pay, English.

Not only that but all auctions are fully e¢ cient � the buyer whovalues the object the most winds up buying it.

Here total surplus is simply the highest value:E [maxfvig] = N/(N + 1).Total surplus=buyers�surplus + seller�s surplus.Implies buyers�surpluses are the same.

If a seller wants to maximize revenue, he can simply use anappropriate minimum bid in any of the designs.

Problems happen if: asymmetry, risk aversion, common values, sellerinfo.


Revenue Equivalence: General

Two mechanisms that

1 yield the same allocation of the object.2 have the same surplus for the bidder with the lowest valuation

Have the following properties

1 The seller�s expected revenue is the same.2 A bidder�s expected surplus is the same.3 Given a bidder�s value, his expected surplus is the same.

Assumptions needed are risk neutrality, private and independentvalues. (Not Symmetry!)


Documents

Microeconomics MSc - Haifatodd/beamerauctionshandout.pdf · Microeconomics MSc Auctions Todd R. Kaplan University of Haifa December 2009 ... One™s bid does not a⁄ect the price